How this instrument works
A circle is completely determined by a single number, its radius, and every other basic measure follows from that one figure through fixed relationships involving π: area = πr², circumference = 2πr, and diameter = 2r. Unlike a rectangle or a triangle, which each need at least two independent measurements to pin down fully, a circle's perfect symmetry collapses everything down to one.
This calculator returns all three measures from a single radius input, useful whenever more than one figure is needed at once — sizing a circular table (diameter for clearance, circumference for edging trim, area for a tablecloth) is a typical case where all three numbers matter for the same physical object.
The three formulas relate to each other in a fixed, unchanging way regardless of the radius entered: circumference is always exactly π times the diameter (since circumference = 2πr = πd), and area is always exactly π times the square of the radius, never linearly related to circumference despite both scaling from the same radius.
- Enter the circle's radius into the Radius field.
- Read Area, Circumference, and Diameter: the sheet computes all three from that one number simultaneously.
- For working backward from a measured circumference to the radius, use the dedicated Circumference Calculator elsewhere on this site.
Worked example — a radius of 5
A circle has a radius of 5 units. Its area is π × 5² = 25π ≈ 78.54 square units, its circumference is 2π × 5 = 10π ≈ 31.42 units, and its diameter is simply 2 × 5 = 10 units — all three figures returned from the single radius, with no additional input needed for any of them.
A smaller circle with radius 3 gives area 9π ≈ 28.27, circumference 6π ≈ 18.85, and diameter 6 — every measure shrinking together as the radius shrinks, though at different rates: area shrinks with the square of the radius while circumference and diameter shrink in direct proportion to it.
Questions
What is the formula for a circle's area?
A = πr², where r is the radius. Area grows with the SQUARE of the radius, so doubling the radius quadruples the area, a much faster growth rate than either the circumference or the diameter, both of which merely double.
How is circumference related to diameter?
Circumference is always exactly π times the diameter: C = πd. This is actually the original definition of π itself — the ratio of any circle's distance around to its distance across, a constant true for every circle regardless of size.
Can I find the radius if I only know the area?
Yes — rearrange A = πr² to r = √(A ⁄ π). This calculator takes the radius as the starting input, but the same three formulas work equally well solved in reverse from any one of the three outputs.
What if the radius is zero?
The circle has collapsed to a single point, and area, circumference, and diameter are all exactly zero — the formulas handle this limit cleanly, with no special case needed.
Why does area use π but scale differently from circumference?
Both formulas share the same constant π, but area's r² term makes it grow quadratically while circumference's plain r term makes it grow linearly — the same underlying geometry (a circle's fixed shape) producing two genuinely different growth rates as the radius changes.